Hecke character

In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function. In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class ofL-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function. A name sometimes used for Hecke character is the German term Größencharakter (often written Grössencharakter, Grossencharacter, etc.). A Hecke character is a character of the idele class group of a number field or global function field. It corresponds uniquely to a character of the idele group which is trivial on principal ideles, via composition with the projection map. This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a 'quasicharacter'), or as a homomorphism to the unit circle in C ('unitary'). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions. The conductor of a Hecke character χ is the largest ideal m such that χ is a Hecke character mod m. Here we say that χ is a Hecke character mod m if χ (considered as a character on the idele group) is trivial on the group of finite ideles whose every v-adic component lies in 1 + mOv. The original definition of a Hecke character, going back to Hecke, was in terms ofa character on fractional ideals. For a number field K, letm = mfm∞ be aK-modulus, with mf, the 'finite part', being an integral ideal of K and m∞, the 'infinite part', being a (formal) product of real places of K. Let Imdenote the group of fractional ideals of K relatively prime to mf andlet Pm denote the subgroup of principal fractional ideals (a)where a is near 1 at each place of m in accordance with the multiplicities ofits factors: for each finite place v in mf, ordv(a − 1) is at least as large as the exponent for v in mf, and a is positive under each real embedding in m∞. A Hecke character with modulus mis a group homomorphism from Im into the nonzero complex numberssuch that on ideals (a) in Pm its value is equal to thevalue at a of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all Archimedean completions of K where each local component of the homomorphism has the same real part (in the exponent). (Here we embed a into the product of Archimedean completions of K using embeddings corresponding to the various Archimedean places on K.) Thus a Hecke character may be defined on the ray class group modulo m, which is the quotient Im/Pm. Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked with moduli where all real places appeared.The role of the infinite part m∞ is now subsumed under the notion ofan infinity-type. The ideal definition is much more complicated than the idelic one, and Hecke's motivation for his definition was to construct L-functions (sometimes referred to as Hecke L-functions) that extend the notion of a Dirichlet L-function from the rationals to other number fields. For a Hecke character χ, its L-function is defined to be the Dirichlet series carried out over integral ideals relatively prime to the modulus m of the Hecke character.The notation N(I) means the ideal norm. The common real part condition governing the behavior of Hecke characters on the subgroups Pm implies theseDirichlet series are absolutely convergent in some right half-plane. Hecke proved these L-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at s = 1 when the character is trivial. For primitive Hecke characters (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these L-functions satisfy a functional equation relating the values of the L-function of a character and the L-function of its complex conjugate character.